Gravitation
Inside A Uniform Hollow Sphere

The gravitational
force inside a hollow sphere shell of uniform areal mass density is everywhere
equal to zero, and may be proved by the following argument:

Let the sphere have
a radius a. Place a point P inside the sphere at a distance r from the
center where r < a; i.e., r is strictly less than a. Draw a line through
P to intersect the sphere at two opposite points. Call these points
and. Let the distance
from P to be
r1, and the distance from P to
be r2.

Now place a differential
area dA
at , and project
straight lines through P to acquire its image dA
at . These two areas
subtend a solid angle d
at P. Let the sphere have areal mass density (kg/m2).
Then the net differential attraction dF of dA
and dA
at P directed toward
is just